Question 1

Solve the problem. -Sketch a continuous curve y = f(x) with the following properties: f(2) = 3; f (x) > 0 for x > 4; and f (x) < 0 for x < 4 .

Accepted Answer

The answer of Solve the problem.
-Sketch a continuous curve y...

Question 2

Determine all critical points for the function.
-$$f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 2$$

Accepted Answer

A)  x = 0 and x = 1 
B)  x = 0 and x = 2 
C)  x = -1 and x = 1 
D)  x = 0 
A)  x = 0 and x = 1 
B)  x = 0 and x = 2 
C)  x = -1 and x = 1 
D)  x = 0

Question 3

Find the largest open interval where the function is changing as requested.
-Increasing $f ( x ) = x ^ { 2 } - 2 x + 1$

Accepted Answer

A)  $( - \infty , 1 )$ 
B)  $( 1 , \infty )$ 
C)  $( 0 , \infty )$ 
D)  $( - \infty , 0 )$ 
A)  $( - \infty , 1 )$ 
B)  $( 1 , \infty )$ 
C)  $( 0 , \infty )$ 
D)  $( - \infty , 0 )$

Question 4

Find all possible functions with the given derivative.
-$y ^ { \prime } = \frac { 7 } { 2 \sqrt { t } }$

Accepted Answer

A)  $\sqrt { t } + C$ 
B)  $\frac { 7 t } { 2 } + C$ 
C)  $\frac { 7 t ^ { 2 } } { 2 } + C$ 
D)  $7 \sqrt { t } + C$ 
A)  $\sqrt { t } + C$ 
B)  $\frac { 7 t } { 2 } + C$ 
C)  $\frac { 7 t ^ { 2 } } { 2 } + C$ 
D)  $7 \sqrt { t } + C$

Question 5

Solve the problem.
-Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's positior $\mathrm { t }$.
$v = \cos \frac { \pi } { 2 } t , s ( 0 ) = 1$

Accepted Answer

A)  $s = \frac { 2 } { \pi } \sin \frac { \pi } { 2 } t + \pi$ 
B)  $s = 2 \pi \sin \frac { \pi } { 2 } t$ 
C)  $s = \frac { 2 } { \pi } \sin \frac { \pi } { 2 } t + 1$ 
D)  $s = \sin t + 1$ time 
A)  $s = \frac { 2 } { \pi } \sin \frac { \pi } { 2 } t + \pi$ 
B)  $s = 2 \pi \sin \frac { \pi } { 2 } t$ 
C)  $s = \frac { 2 } { \pi } \sin \frac { \pi } { 2 } t + 1$ 
D)  $s = \sin t + 1$ time

Question 6

Find the location of the indicated absolute extremum for the function.
-

Accepted Answer

A)  x = -1 
B)  x = 1 
C)  x = 2 
D)  x = -2 
A)  x = -1 
B)  x = 1 
C)  x = 2 
D)  x = -2

Question 7

Find the extreme values of the function and where they occur.
-$$y = x ^ { 2 } e ^ { x }$$

Accepted Answer

A)  Minimum value is 0 at $x = 0$; no maximum value. 
B)  Minimum value is 0 at $x = 0$, maximum value is $4 e ^ { - 2 }$ at $x = - 2$. 
C)  Minimum value is $4 e ^ { - 2 }$ at $x = - 2$; no maximum value. 
D)  None 
A)  Minimum value is 0 at $x = 0$; no maximum value. 
B)  Minimum value is 0 at $x = 0$, maximum value is $4 e ^ { - 2 }$ at $x = - 2$. 
C)  Minimum value is $4 e ^ { - 2 }$ at $x = - 2$; no maximum value. 
D)  None

Question 8

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema.
-$$f ( x ) = 0.1 x ^ { 3 } - 15 x ^ { 2 } - 39 x - 85$$

Accepted Answer

A)  Approximate local maximum at -1.284; approximate local minimum at 101.284 
B)  Approximate local minimum at -101.284; approximate local maximum at 1.284 
C)  Approximate local maximum at -101.284; approximate local minimum at 1.284 
D)  Approximate local minimum at -1.284; approximate local maximum at 101.284 
A)  Approximate local maximum at -1.284; approximate local minimum at 101.284 
B)  Approximate local minimum at -101.284; approximate local maximum at 1.284 
C)  Approximate local maximum at -101.284; approximate local minimum at 1.284 
D)  Approximate local minimum at -1.284; approximate local maximum at 101.284

Question 9

Show that the function has exactly one zero in the given interval.
-$r ( \theta ) = 4 \cot \theta + \frac { 1 } { \theta ^ { 2 } } + 6 , ( 0 , \pi )$

Accepted Answer

The function @#LAT-DLM& is continuous on the open

Question 10

Solve the problem.
-Find the graph that matches the given table.

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 11

Identify the function's local and absolute extreme values, if any, saying where they occur.
-$$f ( r ) = ( r - 10 ) ^ { 3 }$$

Accepted Answer

A)  local minimum: x = 0 
B)  no local extrema 
C)  local minimum: x = 10 
D)  local minimum: x = 0; local maximum: x = 10 
A)  local minimum: x = 0 
B)  no local extrema 
C)  local minimum: x = 10 
D)  local minimum: x = 0; local maximum: x = 10

Question 12

Find the absolute extreme values of the function on the interval.
-$$f ( x ) = \csc x , - \frac { \pi } { 2 } \leq x \leq \frac { 3 \pi } { 2 }$$

Accepted Answer

A)  absolute maximum is 1 at $x = \pi$; absolute minimum is $- 1$ at $x = \pi$ 
B)  absolute maximum is $- 1$ at $x = \pi$; absolute minimum is 1 at $x = 0$ 
C)  absolute maximum is 0 at $x = - \pi$; absolute minimum is $- 1$ at $x = \pi$ 
D)  absolute maximum does not exist; absolute minimum does not exist 
A)  absolute maximum is 1 at $x = \pi$; absolute minimum is $- 1$ at $x = \pi$ 
B)  absolute maximum is $- 1$ at $x = \pi$; absolute minimum is 1 at $x = 0$ 
C)  absolute maximum is 0 at $x = - \pi$; absolute minimum is $- 1$ at $x = \pi$ 
D)  absolute maximum does not exist; absolute minimum does not exist

Question 13

L'Hopital's rule does not help with the given limit. Find the limit some other way.
-$\lim _ { \theta \rightarrow \pi / 2 ^ { - } } \frac { \csc \theta } { \cot \theta }$

Accepted Answer

A)  1 
B)  $\infty$ 
C)  0 
D)  $- 1$ 
A)  1 
B)  $\infty$ 
C)  0 
D)  $- 1$

Question 14

Provide an appropriate response.
-$$\text { Determine the values of constants a and b so that } f ( x ) = a x ^ { 2 } + b x \text { has an absolute maximum at the point } ( 2,4 ) \text {. }$$

Accepted Answer

A)  a = 1, b = 4 
B)  a = 1, b = 2 
C)  a = -1, b = 2 
D)  a = -1, b = 4 
A)  a = 1, b = 4 
B)  a = 1, b = 2 
C)  a = -1, b = 2 
D)  a = -1, b = 4

Question 15

Solve the initial value problem.
-$$\frac { d y } { d x } = \frac { 1 } { 2 \sqrt { x } } + 7 , y ( 4 ) = - 5$$

Accepted Answer

A)  $y = \sqrt { x } + 7 x - 35$ 
B)  $y = \frac { 1 } { \sqrt { x } } + 7 x - \frac { 67 } { 2 }$ 
C)  $y = \sqrt { x } + 7 x + 25$ 
D)  $y = \frac { - 1 } { 4 } \sqrt { x } + 7 x - \frac { 65 } { 2 }$ 
A)  $y = \sqrt { x } + 7 x - 35$ 
B)  $y = \frac { 1 } { \sqrt { x } } + 7 x - \frac { 67 } { 2 }$ 
C)  $y = \sqrt { x } + 7 x + 25$ 
D)  $y = \frac { - 1 } { 4 } \sqrt { x } + 7 x - \frac { 65 } { 2 }$

Question 16

Find the location of the indicated absolute extremum for the function.
-

Accepted Answer

A)  x = 0 
B)  No maximum 
C)  x = 3 
D)  x = 5 
A)  x = 0 
B)  No maximum 
C)  x = 3 
D)  x = 5

Question 17

L'Hopital's rule does not help with the given limit. Find the limit some other way.
-$\lim _ { x \rightarrow \theta ^ { + } } \frac { \sec x } { \tan x }$

Accepted Answer

A)  1 
B)  0 
C)  $\infty$ 
D)  $- 1$ 
A)  1 
B)  0 
C)  $\infty$ 
D)  $- 1$

Question 18

Solve the problem.
-Sketch a smooth curve through the origin with the following properties: $\mathrm { f } ^ { \prime } ( \mathrm { x } ) > 0$ for $\mathrm { x }  0$; $\mathrm { f } ^ { \prime \prime } ( \mathrm { x } )$ approaches 0 as $x$ approaches $- \infty$; and $f ^ { \prime \prime } ( x )$ approaches 0 as $x$ approaches $\infty$.

Accepted Answer

The answer of Solve the problem.
-Sketch a smooth curve through...

Question 19

Solve the problem.
-You are driving along a highway at a steady $86 \mathrm { ft } / \mathrm { sec }$ when you see a deer ahead and slam on the brakes. What constant deceleration is required to stop your car in $264 \mathrm { ft }$ ?

Accepted Answer

A)  $14.01 \mathrm { ft } / \mathrm { sec } ^ { 2 }$ 
B)  $28.02 \mathrm { ft } / \mathrm { sec } ^ { 2 }$ 
C)  $7.00 \mathrm { ft } / \mathrm { sec } ^ { 2 }$ 
D)  $0.07 \mathrm { ft } / \mathrm { sec } ^ { 2 }$ 
A)  $14.01 \mathrm { ft } / \mathrm { sec } ^ { 2 }$ 
B)  $28.02 \mathrm { ft } / \mathrm { sec } ^ { 2 }$ 
C)  $7.00 \mathrm { ft } / \mathrm { sec } ^ { 2 }$ 
D)  $0.07 \mathrm { ft } / \mathrm { sec } ^ { 2 }$

Question 20

Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up
and concave down.
-

Accepted Answer

A)  Local maximum at $x = 3$; local minimum at $x = - 3$; concave up on $( 0 , - 3 )$ and $( 3 , \infty )$; concave down on $( - 3$, 3) 
B)  Local minimum at $x = 3$; local maximum at $x = - 3$; concave up on $( 0 , \infty )$; concave down on $( - \infty , 0 )$ 
C)  Local minimum at $x = 3$; local maximum at $x = - 3 ;$ concave up on $( 0 , - 3 )$ and $( 3 , \infty )$; concave down on $( - 3$, 3) 
D)  Local minimum at $x = 3$; local maximum at $x = - 3 ;$ concave down on $( 0 , \infty )$; concave up on $( - \infty , 0 )$ 
A)  Local maximum at $x = 3$; local minimum at $x = - 3$; concave up on $( 0 , - 3 )$ and $( 3 , \infty )$; concave down on $( - 3$, 3) 
B)  Local minimum at $x = 3$; local maximum at $x = - 3$; concave up on $( 0 , \infty )$; concave down on $( - \infty , 0 )$ 
C)  Local minimum at $x = 3$; local maximum at $x = - 3 ;$ concave up on $( 0 , - 3 )$ and $( 3 , \infty )$; concave down on $( - 3$, 3) 
D)  Local minimum at $x = 3$; local maximum at $x = - 3 ;$ concave down on $( 0 , \infty )$; concave up on $( - \infty , 0 )$

Solve the problem. -Sketch a continuous curve y = f(x) with the following properties: f(2) = 3; f (x) > 0 for x > 4; and f (x) < 0 for x < 4 .

Determine all critical points for the function. - $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 2$

Find the largest open interval where the function is changing as requested. -Increasing $f ( x ) = x ^ { 2 } - 2 x + 1$

Find all possible functions with the given derivative. - $y ^ { \prime } = \frac { 7 } { 2 \sqrt { t } }$

Solve the problem. -Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's positior $\mathrm { t }$ . $v = \cos \frac { \pi } { 2 } t , s ( 0 ) = 1$

Find the location of the indicated absolute extremum for the function. -

Find the extreme values of the function and where they occur. - $y = x ^ { 2 } e ^ { x }$

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f ( x ) = 0.1 x ^ { 3 } - 15 x ^ { 2 } - 39 x - 85$

Show that the function has exactly one zero in the given interval. - $r ( \theta ) = 4 \cot \theta + \frac { 1 } { \theta ^ { 2 } } + 6 , ( 0 , \pi )$

Solve the problem. -Find the graph that matches the given table.

Identify the function's local and absolute extreme values, if any, saying where they occur. - $f ( r ) = ( r - 10 ) ^ { 3 }$

Find the absolute extreme values of the function on the interval. - $f ( x ) = \csc x , - \frac { \pi } { 2 } \leq x \leq \frac { 3 \pi } { 2 }$

L'Hopital's rule does not help with the given limit. Find the limit some other way. - $\lim _ { \theta \rightarrow \pi / 2 ^ { - } } \frac { \csc \theta } { \cot \theta }$

Provide an appropriate response. - $\text { Determine the values of constants a and b so that } f ( x ) = a x ^ { 2 } + b x \text { has an absolute maximum at the point } ( 2,4 ) \text {. }$

Solve the initial value problem. - $\frac { d y } { d x } = \frac { 1 } { 2 \sqrt { x } } + 7 , y ( 4 ) = - 5$

Find the location of the indicated absolute extremum for the function. -

L'Hopital's rule does not help with the given limit. Find the limit some other way. - $\lim _ { x \rightarrow \theta ^ { + } } \frac { \sec x } { \tan x }$

Solve the problem. -You are driving along a highway at a steady $86 \mathrm { ft } / \mathrm { sec }$ when you see a deer ahead and slam on the brakes. What constant deceleration is required to stop your car in $264 \mathrm { ft }$ ?

Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down. -

Functions

Limits and Continuity

Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Exam 5: Applications of Derivatives

Solve the problem. -Sketch a continuous curve y = f(x) with the following properties: f(2) = 3; f (x) > 0 for x > 4; and f (x) < 0 for x < 4 .

Determine all critical points for the function. - f(x)=x3−3x2+2f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 2f(x)=x3−3x2+2

Find the largest open interval where the function is changing as requested. -Increasing f(x)=x2−2x+1f ( x ) = x ^ { 2 } - 2 x + 1f(x)=x2−2x+1

Find all possible functions with the given derivative. - y′=72ty ^ { \prime } = \frac { 7 } { 2 \sqrt { t } }y′=2t​7​

Solve the problem. -Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's positior t\mathrm { t }t . v=cos⁡π2t,s(0)=1v = \cos \frac { \pi } { 2 } t , s ( 0 ) = 1v=cos2π​t,s(0)=1

Find the location of the indicated absolute extremum for the function. -

Find the extreme values of the function and where they occur. - y=x2exy = x ^ { 2 } e ^ { x }y=x2ex

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - f(x)=0.1x3−15x2−39x−85f ( x ) = 0.1 x ^ { 3 } - 15 x ^ { 2 } - 39 x - 85f(x)=0.1x3−15x2−39x−85

Show that the function has exactly one zero in the given interval. - r(θ)=4cot⁡θ+1θ2+6,(0,π)r ( \theta ) = 4 \cot \theta + \frac { 1 } { \theta ^ { 2 } } + 6 , ( 0 , \pi )r(θ)=4cotθ+θ21​+6,(0,π)

Solve the problem. -Find the graph that matches the given table.

Identify the function's local and absolute extreme values, if any, saying where they occur. - f(r)=(r−10)3f ( r ) = ( r - 10 ) ^ { 3 }f(r)=(r−10)3

Find the absolute extreme values of the function on the interval. - f(x)=csc⁡x,−π2≤x≤3π2f ( x ) = \csc x , - \frac { \pi } { 2 } \leq x \leq \frac { 3 \pi } { 2 }f(x)=cscx,−2π​≤x≤23π​

L'Hopital's rule does not help with the given limit. Find the limit some other way. - lim⁡θ→π/2−csc⁡θcot⁡θ\lim _ { \theta \rightarrow \pi / 2 ^ { - } } \frac { \csc \theta } { \cot \theta }limθ→π/2−​cotθcscθ​

Solve the initial value problem. - dydx=12x+7,y(4)=−5\frac { d y } { d x } = \frac { 1 } { 2 \sqrt { x } } + 7 , y ( 4 ) = - 5dxdy​=2x​1​+7,y(4)=−5

Find the location of the indicated absolute extremum for the function. -

L'Hopital's rule does not help with the given limit. Find the limit some other way. - lim⁡x→θ+sec⁡xtan⁡x\lim _ { x \rightarrow \theta ^ { + } } \frac { \sec x } { \tan x }limx→θ+​tanxsecx​

Solve the problem. -You are driving along a highway at a steady 86ft/sec86 \mathrm { ft } / \mathrm { sec }86ft/sec when you see a deer ahead and slam on the brakes. What constant deceleration is required to stop your car in 264ft264 \mathrm { ft }264ft ?

Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down. -

Functions

Limits and Continuity

Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Determine all critical points for the function. - $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 2$

Find the largest open interval where the function is changing as requested. -Increasing $f ( x ) = x ^ { 2 } - 2 x + 1$

Find all possible functions with the given derivative. - $y ^ { \prime } = \frac { 7 } { 2 \sqrt { t } }$

Solve the problem. -Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's positior $\mathrm { t }$ . $v = \cos \frac { \pi } { 2 } t , s ( 0 ) = 1$

Find the extreme values of the function and where they occur. - $y = x ^ { 2 } e ^ { x }$

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f ( x ) = 0.1 x ^ { 3 } - 15 x ^ { 2 } - 39 x - 85$

Show that the function has exactly one zero in the given interval. - $r ( \theta ) = 4 \cot \theta + \frac { 1 } { \theta ^ { 2 } } + 6 , ( 0 , \pi )$

Identify the function's local and absolute extreme values, if any, saying where they occur. - $f ( r ) = ( r - 10 ) ^ { 3 }$

Find the absolute extreme values of the function on the interval. - $f ( x ) = \csc x , - \frac { \pi } { 2 } \leq x \leq \frac { 3 \pi } { 2 }$

L'Hopital's rule does not help with the given limit. Find the limit some other way. - $\lim _ { \theta \rightarrow \pi / 2 ^ { - } } \frac { \csc \theta } { \cot \theta }$

Solve the initial value problem. - $\frac { d y } { d x } = \frac { 1 } { 2 \sqrt { x } } + 7 , y ( 4 ) = - 5$

L'Hopital's rule does not help with the given limit. Find the limit some other way. - $\lim _ { x \rightarrow \theta ^ { + } } \frac { \sec x } { \tan x }$

Solve the problem. -You are driving along a highway at a steady $86 \mathrm { ft } / \mathrm { sec }$ when you see a deer ahead and slam on the brakes. What constant deceleration is required to stop your car in $264 \mathrm { ft }$ ?